Three-dimensional stress and strain distribution in a two-layer model of a coronary artery

For a right coronary artery, three-dimensional stress and strain distributions at a physiological intraluminal pressure and an axial extension ratio were computed on the basis of a two-layer elastic model. To validate the model, curves of external radius versus pressure and of axial force versus pressure were computed for three axial extension ratios. To analyze mechanical properties, stress-free configurations of media and adventitia, and the constitutive law of each layer in literature, were used. The present study showed that the peak circumferential stress and the peak axial stress appear in the media at the boundary between the media and adventitia. This result is due to the opening angle of the media being larger than π (rad) and the larger value of a material constant of the strain energy function for the media than for the adventitia. The circumferential stress and strain were discontinuous at the boundary. On the other hand, the radial stress was continuous at the boundary because of the boundary condition for stress. The circumferential stress and axial stress in the adventitia were almost uniformly distributed, and smaller than in the media. The residual stress and strain were also computed. The circumferential residual stress and strain were almost linearly distributed in each layer, although discontinuity appeared at the boundary between the two layers.     

Active axial stress in mouse aorta

The study verifies the development of active axial stress in the wall of mouse aorta over a range of physiological loads when the smooth muscle cells are stimulated to contract. The results obtained show that the active axial stress is virtually independent of the magnitude of pressure, but depends predominately on the longitudinal stretch ratio. The dependence is non-monotonic and is similar to the active stress-stretch dependence in the circumferential direction reported in the literature. The expression for the active axial stress fitted to the experimental data shows that the maximum active stress is developed at longitudinal stretch ratio 1.81, and 1.56 is the longitudinal stretch ratio below which the stimulation does not generate active stress. The study shows that the magnitude of active axial stress is smaller than the active circumferential stress. There is need for more experimental investigations on the active response of different types of arteries from different species and pathological conditions. The results of these studies can promote building of refined constrictive models in vascular rheology.     

Determining the mechanical properties of yeast cell walls

The intrinsic cell wall mechanical properties of Baker's yeast (Saccharomyces cerevisiae) cells were determined. Force-deformation data from compression of individual cells up to failure were recorded, and these data were fitted by an analytical model to extract the elastic modulus of the cell wall and the initial stretch ratio of the cell. The cell wall was assumed to be homogeneous, isotropic, and incompressible. A linear elastic constitutive equation was assumed based on Hencky strains to accommodate the large stretches of the cell wall. Because of the high compression speed, water loss during compression could be assumed to be negligible. It was then possible to treat the initial stretch ratio and elastic modulus as adjustable parameters within the analytical model. As the experimental data fitted numerical simulations well up to the point of cell rupture, it was also possible to extract cell wall failure criteria. The mean cell wall properties for resuspended dried Baker's yeast were as follows: elastic modulus 185 ± 15 MPa, initial stretch ratio 1.039 ± 0.006, circumferential stress at failure 115 ± 5 MPa, circumferential strain at failure 0.46 ± 0.03, and strain energy per unit volume at failure 30 ± 3 MPa. Data on yeast cells obtained by this method and model should be useful in the design and optimization of cell disruption equipment for yeast cell processing.     

Contribution of tree structures in the lung to lung elastic recoil

The direct contribution of forces in tree structures in the lung to lung recoil pressure and changes in recoil pressure induced by alterations of the forces are analyzed. The analysis distinguishes the contributions of axial and circumferential tensions in the trees and indicates that only axial tensions directly contribute to static recoil. This contribution is derived from analysis of the axial forces transmitted across a random plane transecting the lung. The change in recoil pressure induced by changes in axial tension is similarly derived. Alterations of circumferential tensions in the trees indirectly change recoil by causing nonuniform deformations of the surrounding lung parenchyma, and a continuum elasticity solution for the stress induced by the deformations is derived. Sample calculations are presented for the airway tree based on available data on airway morphometric and mechanical properties. The increase in recoil pressure accompanying increases in axial and circumferential tensions with contraction of airway smooth muscle is also analyzed. The calculations indicate that axial stresses in the airway tree out to bronchioles directly contribute only a small fraction of the static recoil pressure. However, it is found that contraction of smooth muscle in these airways can increase recoil pressure appreciably (10-20%), mainly by the deformation of the parenchyma with increases in circumferential tension in smaller airways. The results indicate that the geometric and mechanical properties of the airway tree are such that only peripheral elements of the tree can substantially affect the elastic properties of the lung. The possible contributions of vascular trees for which data on mechanical and morphometric properties are more limited are also discussed.     

Stress gradients in the walls of large arteries

Elastic behavior of vascular wall, assuming the vessels to be ‘thick-walled’ and utilizing finite deformation theory, was investigated. It was found that canine carotid arterial wall is neither isotropic nor transversely isotropic. Previously, stress-strain relations were obtained for carotid arteries on the basis of membrane theory (Doyle and Dobrin, 1971). Since strain gradients across the wall are fairly steep, the applicability of such expressions, for pointwise evaluation of stress, required examination. The study indicated that these relationships between mean circumferential stress and mean extension ratio in the circumferential direction could be used to relate the specific circumferential stress value to the specific extension ratio at any designated point within the wall. From this analysis it was possible to evaluate circumferential and radial wall stresses. Both of these stresses are maximal at the inner surface of the intima. At this point the radial stress is equal to the transmural pressure and is compressive, while the circumferential stress is tensile and is 1·5 to 2 times the value of the mean stress, i.e. the product of transmural pressure and the ratio of internal radius-to-wall thickness. Both stresses are lowest at the outer edge of the adventitia. These stress distributions were considered with respect to the spacing of the elastic lamellae and the absence of discernible vasa vasora in the inner third of the wall.     

Determination of the axial and circumferential mechanical properties of the skin tissue using experimental testing and constitutive modeling

The skin, being a multi-layered material, is responsible for protecting the human body from the mechanical, bacterial, and viral insults. The skin tissue may display different mechanical properties according to the anatomical locations of a body. However, these mechanical properties in different anatomical regions and at different loading directions (axial and circumferential) of the mice body to date have not been determined. In this study, the axial and circumferential loads were imposed on the mice skin samples. The elastic modulus and maximum stress of the skin tissues were measured before the failure occurred. The nonlinear mechanical behavior of the skin tissues was also computationally investigated through a suitable constitutive equation. Hyperelastic material model was calibrated using the experimental data. Regardless of the anatomic locations of the mice body, the results revealed significantly different mechanical properties in the axial and circumferential directions and, consequently, the mice skin tissue behaves like a pure anisotropic material. The highest elastic modulus was observed in the back skin under the circumferential direction (6.67 MPa), while the lowest one was seen in the abdomen skin under circumferential loading (0.80 MPa). The Ogden material model was narrowly captured the nonlinear mechanical response of the skin at different loading directions. The results help to understand the isotropic/anisotropic mechanical behavior of the skin tissue at different anatomical locations. They also have implications for a diversity of disciplines, i.e., dermatology, cosmetics industry, clinical decision making, and clinical intervention.     

Adaptation of passive rat left ventricle in diastolic dysfunction.

This article deals with providing a theoretical explanation for quantitative changes in the geometry, the opening angle and the deformation parameters of the rat ventricular wall during adaptation of the passive left ventricle in diastolic dysfunction. A large deformation theory is applied to analyse transmural stress and strain distribution in the left ventricular wall considering it to be made of homogeneous, incompressible, transversely isotropic, non-linear elastic material. The basic assumptions made for computing stress distributions are that the average circumferential stress and strain for the adaptive ventricle is equal to the average circumferential stress and strain in the normotensive ventricle, respectively.All the relevant parameters, such as opening angle, twist per unit length, axial extension, internal and external radii and others, in the stress-free, unloaded and loaded states of normotensive, hypertensive and adaptive left ventricle are determined. The circumferential stress and strain distribution through the ventricular wall are also computed. Our analysis predicts that during adaptation, wall thickness and wall mass of the ventricle increase. These results are consistent with experimental findings and are the indications of initiation of congestive heart failure.     

Transmission characteristics of axial waves in blood vessels

The elastic behavior of blood vessels can be quantitatively examined by measuring the propagation characteristics of waves transmitted by them. In addition, specific information regarding the viscoelastic properties of the vessel wall can be deduced by comparing the observed wave transmission data with theoretical predictions. The relevance of these deductions is directly dependent on the validity of the mathematical model for the mechanical behavior of blood vessels used in the theoretical analysis. Previous experimental investigations of waves in blood vessels have been restricted to pressure waves even though theoretical studies predict three types of waves with distinctly different transmission characteristics. These waves can be distinguished by the dominant displacement component of the vessel wall and are accordingly referred to as radial, axial and circumferential waves. The radial waves are also referred to as pressure waves since they exhibit pronounced pressure fluctuations. For a thorough evaluation of the mathematical models used in the analysis it is necessary to measure also the dispersion and attenuation of the axial and circumferential (torsion) waves.

To this end a method has been developed to determine the phase velocities and damping of sinusoidal axial waves in the carotid artery of anesthetized dogs with the aid of an electro-optical tracking system. For frequencies between 25 and 150 Hz the speed of the axial waves was between 20 and 40 m/sec and generally increased with frequency, while the natural pressure wave travelled at a speed of about 10 m/sec. On the basis of an isotropic wall model the axial wave speed should however be approximately 5 times higher than the pressure wave speed. This discrepancy can be interpreted as an indication for an anisotropic behavior of the carotid wall. The carotid artery appears to be more elastic in the axial than in the circumferential direction.     

A piece-wise non-linear elastic stress expression of human and pig coronary arteries tested in vitro.

Eight human and nineteen pig unembalmed proximal left anterior descending and circumflex coronary arteries were subjected to linear volume changes (2 s ramp time) at three fixed axial extensions while immersed in a physiological saline bath at body temperature. Measured parameters included: lumen pressure, outside diameter, axial force, and axial extension. The deformations were measured using a video dimensional analyzer. The arteries were inflated to pressures well above the physiological range at each axial extension. A latex inner tube was placed inside of each specimen to prevent leakage, and its effects upon the measured stresses were corrected analytically. With this method, the average circumferential and axial stresses could be computed directly from the experimental data. In both directions the average stresses measured displayed two distinct regions: stresses occurring for small diameter changes (physiological pressures) and stresses occurring for large diameter changes (high pressures). The resulting average small strain and large strain stress components were curve-fit separately and, when reassembled, provided a piece-wise model of the stress response of coronary arteries over a wide range of inflation pressures and axial extensions.     

Pipette aspiration technique for the measurement of nonlinear and anisotropic mechanical properties of blood vessel walls under biaxial stretch

A pipette aspiration technique was proposed for the measurement of nonlinear mechanical properties of arteries under biaxial stretching. A cross-shaped specimen of porcine thoracic aorta whose principal axes corresponded with the axial and circumferential directions of the aortic walls was excised. The intraluminal surface of the specimen was aspirated with a circular cross-sectioned glass pipette while the specimen was stretching in the axial and circumferential directions in 10% increments. The elastic modulus agreed with the incremental elastic modulus obtained through a conventional pressure-diameter test of the same specimen to within an error of 30% at a circumferential stretch ratio below 1.3 and an axial stretch ratio of 1.0, 1.1 or 1.2, which represent lower range of physiological stretch ratios for the porcine aorta. A rectangular cross-sectioned pipette was utilized to measure anisotropic properties of the specimen under biaxial stretching. When aspirated with such a pipette, the specimens' elastic properties along the length of the rectangular pipette cross section can be neglected. The elastic modulus was found to increase rapidly when the specimen was stretched in the direction of the pipette's width. Thus, pipette aspiration should have many advantages such as well measurement of the local nonlinear and anisotropic mechanical properties of blood vessel walls.     

Mechanical anisotropy of inflated elastic tissue from the pig aorta

Uniaxial and biaxial mechanical properties of purified elastic tissue from the proximal thoracic aorta were studied to understand physiological load distributions within the arterial wall. Stress–strain behaviour was non-linear in uniaxial and inflation tests. Elastic tissue was 40% stiffer in the circumferential direction compared to axial in uniaxial tests and～100% stiffer in vessels at an axial stretch ratio of 1.2 or 1.3 and inflated to physiological pressure. Poisson’s ratio v_θz averaged 0.2 and v_zθ increased with circumferential stretch from ～0.2 to ～0.4. Axial stretch had little impact on circumferential behaviour. In intact (unpurified) vessels at constant length, axial forces decreased with pressure at low axial stretches but remained constant at higher stretches. Such a constant axial force is characteristic of incrementally isotropic arteries at their in vivo dimensions. In purified elastic tissue, force decreased with pressure at all axial strains, showing no trend towards isotropy. Analysis of the force–length–pressure data indicated a vessel with v_θz≈0.2 would stretch axially 2–4% with the cardiac pulse yet maintain constant axial force. We compared the ability of 4 mathematical models to predict the pressure-circumferential stretch behaviour of tethered, purified elastic tissue. Models that assumed isotropy could not predict the stretch at zero pressure. The neo-Hookean model overestimated the non-linearity of the response and two non-linear models underestimated it. A model incorporating contributions from orthogonal fibres captured the non-linearity but not the zero-pressure response. Models incorporating anisotropy and non-linearity should better predict the mechanical behaviour of elastic tissue of the proximal thoracic aorta.     

Effects of a sustained extension on arterial growth and remodeling: a theoretical study

Three recent studies reveal that the unloaded length of a carotid artery increases significantly and rapidly in response to sustained increases in axial extension. Moreover, such lengthening involves an "unprecedented" increase in the rate of turnover of cells and matrix. Although current data are not sufficient for detailed biomechanical analyses, we present general numerical simulations that are consistent with the reported observations and support the hypothesis that rates of turnover correlate with the extent that stresses are perturbed from normal. In particular, a 3-D analysis of wall stress suggests that moderate (15%) increases in axial extension can increase the axial stress to a much greater extent than marked (50%) increases in blood pressure increase the circumferential stress. Furthermore, such increases in axial stress can occur without inducing significant gradients in stress within the wall. Consequently, we use a new, 2-D constrained mixture model to study evolving changes in the geometry, structure, and properties of carotid arteries in response to a sustained increase in axial extension. These simulations are qualitatively similar to the reports in the literature and support the notion that the stress-free lengths of individual constituents evolve during growth and remodeling.     

Axial prestretch and circumferential distensibility in biomechanics of abdominal aorta

Elastic arteries are significantly prestretched in an axial direction. This property minimises axial deformations during pressure cycle. Ageing-induced changes in arterial biomechanics, among others, are manifested via a marked decrease in the prestretch. Although this fact is well known, little attention has been paid to the effect of decreased prestretch on mechanical response. Our study presents the results of an analytical simulation of the inflation–extension behaviour of the human abdominal aorta treated as nonlinear, anisotropic, prestrained thin-walled as well as thick-walled tube with closed ends. The constitutive parameters and geometries for 17 aortas adopted from the literature were supplemented with initial axial prestretches obtained from the statistics of 365 autopsy measurements. For each aorta, the inflation–extension response was calculated three times, with the expected value of the initial prestretch and with the upper and lower confidence limit of the initial prestretch derived from the statistics. This approach enabled age-related trends to be evaluated bearing in mind the uncertainty in the prestretch. Despite significantly decreased longitudinal prestretch with age, the biomechanical response of human abdominal aorta changes substantially depending on the initial axial stretch was used. In particular, substituting the upper limit of initial prestretch gave mechanical responses which can be characterised by (1) low variation in axial stretch and (2) high circumferential distensibility during pressurisation, in contrast to the responses obtained for their weakly prestretched counterparts. The simulation also suggested the significant effect of the axial prestretch on the variation of axial stress in the pressure cycle. Finally, the obtained results are in accordance with the hypothesis that circumferential-to-axial stiffness ratio is the quantity relatively constant within this cycle.     

Significant Impact on Muscle Mechanics of Small Nonlinearities in Myofilament Elasticity

Important mechanisms in muscle contraction have recently been reevaluated based on analyses that rely on the assumption of linear myofilament elasticity. However, the present theoretical study shows that nonlinearity of this elasticity, even when so minor that it may be difficult to detect in experimental data, could have great impact on the interpretation of muscle mechanical experiments. This is illustrated by using simulated stiffness and strain-versus-force data for muscle fibers shortening at different constant velocities. There is substantial quantitative agreement, for this condition, between models with distributed myofilament compliance and models where the compliance of the myofilaments and the actomyosin cross-bridges are lumped together into two separate elastic elements acting in series. The data thus support the usefulness of the latter, simpler, type of model in the analysis. However, most importantly, the data emphasize the importance of caution before reevaluating fundamental mechanisms of muscle contraction based on analyses relying on the assumption of linear myofilament elasticity.     

Flow-induced shear strain in intima of porcine coronary arteries.

The in vivo circumferential strain has a small variation throughout the vascular system (aorta to arterioles). The axial strain has also been shown to be nearly the same as the circumferential strain under physiological loading. Since the endothelium is mechanically much softer than the media-adventitia in healthy arteries, the porcine intima was considered as a mechanically distinct layer from the media-adventitia in a two-layer computational model. Based on the simulation result, we hypothesize that the flow-induced shear strain in intima can be of similar value as the pressure-induced circumferential strain in healthy coronary arteries, even though the shear stress is orders of magnitude smaller than the circumferential stress. The nearly isotropic deformation (circumferential, axial, and shear strains) may have important implications for mechanical homeostasis of endothelial cells, mechanotransduction, growth, and remodeling of blood vessels.     

An experimental study on the ultimate strength of the adventitia and media of human atherosclerotic carotid arteries in circumferential and axial directions

Atherosclerotic plaque may rupture without warning causing heart attack or stroke. Knowledge of the ultimate strength of human atherosclerotic tissues is essential for understanding the rupture mechanism and predicting cardiovascular events. Despite its great importance, experimental data on ultimate strength of human atherosclerotic carotid artery remains very sparse. This study determined the uniaxial tensile strength of human carotid artery sections containing type II and III lesions (AHA classifications). Axial and circumferential oriented adventitia, media and intact specimens (total=73) were prepared from 6 arteries. The ultimate strength in uniaxial tension was taken as the peak stress recorded when the specimen showed the first evidence of failure and the extensibility was taken as the stretch ratio at failure. The mean adventitia strength values calculated using the first Piola–Kirchoff stress were 1996±867 and 1802±703 kPa in the axial and circumferential directions respectively, while the corresponding values for the media sections were 519±270 and 1230±533 kPa. The intact specimens showed ultimate strengths similar to media in circumferential direction but were twice as strong as the media in the axial direction. Results also indicated that adventitia, media and intact specimens exhibited similar extensibility at failure, in both the axial and circumferential directions (stretch ratio 1.50±0.22). These measurements of the material strength limits for human atherosclerotic carotid arteries could be useful in improving computational models that assess plaque vulnerability.     

Dynamic visco-elastic buckling analysis of an in vitro airway model: prediction of circumferential and axial buckling mode numbers

The purpose of the research reported here was to elucidate the mechanism of formation of the various lobes observed in asthmatic airways by both theoretical and experimental analysis employing an in vitro airway model. The rationale is that the elucidated mechanism will facilitate the development of new diagnostic methods and treatment regimens for asthma. Lobe formation was analyzed on the basis of an assumption of cross-sectional buckling of the airway. Here, we propose a dynamic visco-elastic buckling model analysis of the airway for the prediction of circumferential and axial buckling mode numbers. The calculated circumferential buckling mode numbers were in reasonably good agreement with those measured in the dynamic buckling experiment using the in vitro airway model. The calculated axial buckling mode numbers were in qualitative agreement with those observed in the experiment. The non-dimensional parameters related to the remodeling and the consequent pathologies occurring in asthmatic airways were also shown, and the influence of changes in the non-dimensional parameters on the circumferential and axial buckling mode numbers was also calculated. The circumferential and axial buckling mode numbers decreased due to thickening and stiffening of the basement membrane. Thickening of the tissues surrounding the basement membrane and the increase in the mucus secreted in the airway lumen were modeled as an increase in the added mass on the basement membrane. The results of calculation showed that the circumferential and axial buckling mode numbers increased because of the thickening of the surrounding tissues and the increase in mucus secretion. We suggest that it may be possible to diagnose the severity of asthma by using the results of the calculation of the changes in the buckling mode numbers caused by the changes in the strength of the remodeling. The physiological reality of the in vitro airway model reported here is discussed using the non-dimensional parameters.     

A linearized and incompressible constitutive model for arteries

In many biomechanical studies, blood vessels can be modeled as pseudoelastic orthotropic materials that are incompressible (volume-preserving) under physiological loading. To use a minimum number of elastic constants to describe the constitutive behavior of arteries, we adopt a generalized Hooke's law for the co-rotational Cauchy stress and a recently proposed logarithmic-exponential strain. This strain tensor absorbs the material nonlinearity and its trace is zero for volume-preserving deformations. Thus, the relationships between model parameters due to the incompressibility constraint are easy to analyze and interpret. In particular, the number of independent elastic constants reduces from ten to seven in the orthotropic model. As an illustratory study, we fit this model to measured data of porcine coronary arteries in inflation-stretch tests. Four parameters, n (material nonlinearity), Young's moduli E? (circumferential), E? (axial), and E? (radial) are necessary to fit the data. The advantages and limitations of this model are discussed.     

Loss of stability: a new look at the physics of cell wall behavior during plant cell growth

In this article we investigate aspects of turgor-driven plant cell growth within the framework of a model derived from the Eulerian concept of instability. In particular we explore the relationship between cell geometry and cell turgor pressure by extending loss of stability theory to encompass cylindrical cells. Beginning with an analysis of the three-dimensional stress and strain of a cylindrical pressure vessel, we demonstrate that loss of stability is the inevitable result of gradually increasing internal pressure in a cylindrical cell. The turgor pressure predictions based on this model differ from the more traditional viscoelastic or creep-based models in that they incorporate both cell geometry and wall mechanical properties in a single term. To confirm our predicted working turgor pressures, we obtained wall dimensions, elastic moduli, and turgor pressures of sequential internodal cells of intact Chara corallina plants by direct measurement. The results show that turgor pressure predictions based on loss of stability theory fall within the expected physiological range of turgor pressures for this plant. We also studied the effect of varying wall Poisson's ratio nu on extension growth in living cells, showing that while increasing elastic modulus has an understandably negative effect on wall expansion, increasing Poisson's ratio would be expected to accelerate wall expansion.